Diffusion of hydrogen in niobium in the presence of trapping impurities studied by neutron spectroscopy

The diffusion of hydrogen in niobium with interstitial impurities was investigated by highresolution neutron spectroscopy for the system $\mathrm{Nb}{\mathrm{N}}_x{\mathrm{H}}_y$ with $x=0.4\mathrm{} \mathrm{and} 0.7$ at.% and $y=0.4\mathrm{} \mathrm{and} 0.3$ at.%, respectively. The neutron spectrum at larger scattering vector $\overrightarrow{\mathrm{Q}}$ consists of two parts: a narrow line centered at energy transfer $\hslash \omega =0\mathrm{}$ (width 0.1-3 μeV) which is caused which is caused by hydrogen trapped on nitrogen atoms, and a broad component in the spectrum from hydrogen atoms which diffuse in the more or less undisturbed regions of the lattice. At small $\overrightarrow{\mathrm{Q}}$, the spectral width is directly related to the self-diffusion constant. The experimental spectra, measured as a function of temperature and scattering vector, were interpreted by two theoretical models: (i) a two-state random-walk model (RWM) where the hydrogen alternates between a trapped state and a state of undisturbed diffusion. The RWM is characterized by the mean escape rate from the trap $\frac{1}{{\tau }_0}$, and the capture rate on the traps $\frac{1}{{\tau }_1}$; and (ii) an elastic-continuum model: the nitrogen-hydrogen interaction is treated in terms of the elastic strain field produced by the interstitial nitrogen and hydrogen atoms, and in terms of a short-range hard-core repulsion. This model uses the elastic parameters of the niobium and the interstitial nitrogen and hydrogen atoms. The hard-core radius $r_0$ is the only disposable parameter of the model. The RWM should hold as long as $\frac{2\pi }{\overrightarrow{\mathrm{Q}}}$ is larger than the linear dimensions of the trapping region. Model (i) yields a very good and consistent description of the measured spectra as a function of the concentration $x$, the temperature and the scattering vector. The resulting parameters ${\tau }_0$ and ${\tau }_1$ have the predicted behavior. In particular, ${\tau }_0$ is independent of $x$, and ${\tau }_1^{-1\mathrm{}}$ is proportional to $\mathrm{xD}\left(T\right)\mathrm{}$ where $D$ is the self-diffusion constant in pure niobium. The trapping times are about two orders of magnitude larger than the mean rest time in pure niobium. Using a hard-core radius of $r_0\simeq 2.3$ Å, model (ii) describes very well the experimental spectra at small and large $\overrightarrow{\mathrm{Q}}$ values.